On-shell symmetries

TL;DR

This paper introduces the concept of on-shell symmetries in Lagrangian systems, analyzing how they preserve solution spaces and relate to internal symmetries of PDEs, with implications for theoretical physics.

Contribution

It formally defines on-shell symmetries and explores their properties and constraints within Lagrangian frameworks, linking to internal symmetry theories of PDEs.

Findings

On-shell symmetries strongly constrain variations of the Poincare'-Cartan form.

Solution-preserving transformations are characterized in detail.

Relations between on-shell symmetries and internal PDE symmetries are discussed.

Abstract

We define on-shell symmetries and characterize them for Lagrangian systems. The terms appearing in the variation of the Poincare'-Cartan form, which vanish because of field equations, are found to be strongly constrained if the space of solutions has to be preserved. The behaviour with respect to solution dragging is also investigated in order to discuss relations with the theory of internal symmetries of a PDE.